let rseq1 be Real_Sequence; :: thesis: for seq being Complex_Sequence st ( for n being Element of NAT holds
( seq . n <> 0c & rseq1 . n = (|.seq.| . (n + 1)) / (|.seq.| . n) ) ) & rseq1 is convergent & lim rseq1 < 1 holds
seq is absolutely_summable
let seq be Complex_Sequence; :: thesis: ( ( for n being Element of NAT holds
( seq . n <> 0c & rseq1 . n = (|.seq.| . (n + 1)) / (|.seq.| . n) ) ) & rseq1 is convergent & lim rseq1 < 1 implies seq is absolutely_summable )
assume A1: ( ( for n being Element of NAT holds
( seq . n <> 0c & rseq1 . n = (|.seq.| . (n + 1)) / (|.seq.| . n) ) ) & rseq1 is convergent & lim rseq1 < 1 ) ; :: thesis: seq is absolutely_summable

now

let n be Element of NAT ; :: thesis: ( |.seq.| . n <> 0 & rseq1 . n = (|.seq.| . (n + 1)) / (|.seq.| . n) & |.seq.| . n <> 0 & rseq1 . n = ((abs |.seq.|) . (n + 1)) / (|.seq.| . n) & |.seq.| . n <> 0 & rseq1 . n = ((abs |.seq.|) . (n + 1)) / ((abs |.seq.|) . n) )
A2: ( seq . n <> 0c & rseq1 . n = (|.seq.| . (n + 1)) / (|.seq.| . n) ) by A1;
A3: |.seq.| . n = |.(seq . n).| by VALUED_1:18;
hence ( |.seq.| . n <> 0 & rseq1 . n = (|.seq.| . (n + 1)) / (|.seq.| . n) ) by A2, COMPLEX1:133; :: thesis: ( |.seq.| . n <> 0 & rseq1 . n = ((abs |.seq.|) . (n + 1)) / (|.seq.| . n) & |.seq.| . n <> 0 & rseq1 . n = ((abs |.seq.|) . (n + 1)) / ((abs |.seq.|) . n) )
thus ( |.seq.| . n <> 0 & rseq1 . n = ((abs |.seq.|) . (n + 1)) / (|.seq.| . n) ) by A2, A3, COMPLEX1:133; :: thesis: ( |.seq.| . n <> 0 & rseq1 . n = ((abs |.seq.|) . (n + 1)) / ((abs |.seq.|) . n) )
thus ( |.seq.| . n <> 0 & rseq1 . n = ((abs |.seq.|) . (n + 1)) / ((abs |.seq.|) . n) ) by A2, A3, COMPLEX1:133; :: thesis: verum

end;

then |.seq.| is absolutely_summable by A1, SERIES_1:42;
then abs |.seq.| is summable ;
hence seq is absolutely_summable by Def11; :: thesis: verum